Proof Sketch.
The proof follows from Theorem 1 of [1]}, which proved that the eigendecay of a rotationally invariant kernel \(\kappa (.):[-1,1]\rightarrow \mathbb {R}\)  can be determined based on its asymptotic expansions around the endpoints, \(\pm 1\) . In particular, when
\(\nonumber \kappa (1-t) &=& p_{+1}(t)+c_{+1}t^{\theta } + o(t^{\theta }),\\\nonumber \kappa (-1+t) &=& p_{-1}(t)+c_{-1}t^{\theta } + o(t^{\theta }),\) 
